Efficient Illumination by High Dynamic Range Images

Transcription

1 Eurographics Symposium on Rendering 23 Per Christensen and Daniel Cohen-Or (Editors) Efficient Illumination by High Dynamic Range Images Thomas Kollig 1 and Alexander Keller 2 1 Department of Computer Science, University of Kaiserslautern, Germany 2 Department of Computer Science, University of Ulm, Germany Abstract We present an algorithm for determining uadrature rules for computing the direct illumination of predominantly diffuse objects by high dynamic range images The new method precisely reproduces fine shadow detail, is much more efficient as compared to Monte Carlo integration, and does not reuire any manual intervention 1 Introduction Rendering synthetic objects into real scenes reuires their illumination by real world radiance, which can be captured as a high resolution spherical high dynamic range image 2, 15 This techniue has been introduced by Debevec et al 4, 3, is established in commercial products, and has been used in many movies In this context we address the efficient approximation of the radiance L(x,ω o) L hdr (ω)v (x,ω) f r(ω,x,ω o) n(x),ω dω, Ω + x leaving a point x in direction ω o The spherical high dynamic range image L hdr is of high resolution Ω + x is the upper hemisphere around the surface normal n(x) in point x The visibility V (x,ω) is one if starting in point x no objects are seen in direction ω and zero otherwise For highly directional bidirectional reflectance distribution functions f r it is easy to evaluate the above integral by variance reduced Monte Carlo integration: L(x,ω o) 1 N N 1 L hdr (ω i )V (x,ω i ) f r(ω i,x,ω o) n(x),ω i p(ω i ), (1) where the directions ω i are p-distributed proportional to f r While this is an efficient procedure for highly specular surfaces, ie narrow cones of reflection, it utterly fails for predominantly diffuse surfaces due to high variance intrinsic to L hdr Choosing the directions ω i proportional to L hdr cannot be realized in an efficient way: On the one hand adaptive integration schemes like eg the VEGAS or MISER techniue 11 are inefficient due to excessive additional book keeping data structures On the other hand importance sampling on the discrete set of directional light sources given by the pixels of the high dynamic range image still results in high variance as will be shown in section 5 In addition approximations 7 usually do not perform any better than pure random sampling in this case We also want to avoid variance reduction by manual stratification 3 Projecting the integral operator into the spherical harmonics basis 12, 13 allows for fast rendering However, including detailed shadows in this model imposes considerable cost and restrictions 14 In this paper the above problem of illuminating predominantly diffuse surfaces by high resolution spherical high dynamic range images is addressed Therefore a new method (sections 2 and 3) is presented that reliably captures all details of the high dynamic range image without manual intervention Although seemingly similar to LightGen 1, our techniue yields an almost optimal integration scheme by removing some flaws of the straightforward approach This is demonstrated by numerical evidence Furthermore new superior anti-aliasing techniues are developed (section 4) that reproduce fine shadow details much more precisely and are faster than previous approaches 2 Quadrature Rule Construction The basic idea of our algorithm is to determine a uadrature rule (ω i,b i ) N 1 only depending on the high dynamic range image L hdr Then the direct illumination can be approximated by L(x,ω o) N 1 B i V (x,ω i ) f r(ω i,x,ω o) n(x),ω i, (2) where (ω i,b i ) in fact corresponds to a directional light source from direction ω i with radiosity B i c The Eurographics Association 23

2 Kollig and Keller / Efficient Illumination by High Dynamic Range Images Figure 1: The N = 256 colored points in each image indicate the directions ωi generated by Lloyd s relaxation algorithm on the left and our improved scheme on the right For convenience the spherical images are displayed as 2:1 lattitude/longitude maps Obviously the new approach captures the light distribution much more precisely resulting in a smaller integration error during rendering Both images have been tone mapped for display Figure 2: Images rendered using the uadrature rules generated by Lloyd s relaxation algorithm (top row) and our improved scheme (bottom row) as illustrated in figure 1 The number of light sources is N = 32, 64, 128, 256 (from left to right) The shadow boundary artifacts caused by the directional light sources vanish much faster with our new scheme, clearly indicating the faster decay of the integration error due to the more eualized and conseuently smaller weights of the uadrature rule For a given partition (Ωi )N 1 of the set of all unit directions Ω, the uadrature rule can be determined by Z Bi := Ωi Lhdr (ω)dω (3) and choosing the directions ωi as mass centroids of Ωi, ie Z n 2 ω i ω Ω arccos(hω, ωi) klhdr (ω)kdω Ωi Z = inf ω Ω Ωi o 2 arccos(hω, ωi) klhdr (ω)kdω Thus the uadrature rule is based on the piecewise constant approximation N 1 Lhdr (ω) Bi χ (ω), Ωi Ωi where χωi is the characteristic function of Ωi Ω In the seuel it is shown how to construct a partition such that the partition (Ωi )N 1 is the spherical Voronoi diagram of the set (ωi )N 1 limiting the integration error of (2) if the visibility term V is neglected and max i<n kbi k is as small as possible limiting the integration error of (2) made on each solid angle Ωi Note that by the first constraint an implicit definition of Ωi and ωi is formed and the second one bounds!2 2 Z N 1 N 1 1 N 1 1 Bi N Bi = Bi N Ω Lhdr (ω)dω 3 Determining the Quadrature Rule The algorithm for determining the uadrature rule 5 (ωi, Bi )N 1 is based on Lloyd s relaxation method on the sphere: c The Eurographics Association 23

3 Kollig and Keller / Efficient Illumination by High Dynamic Range Images maximum angle of decentration maximum radiosity (weight) number N of directions number N of directions maximum angle of decentration maximum radiosity (weight) number N of directions number N of directions θ T = arccos(99) θ T = arccos(999) θ T = arccos(9999) Figure 3: Comparison of the uadrature rules generated by Lloyd s relaxation algorithm on the left and our improved scheme on the right using different termination thresholds θ T The top row shows the maximum angle of decentration Hardly noticeable the new scheme obtains a smaller decentration at a somewhat higher variance However, as shown in the bottom row, the decay of the maximum radiosity is uadratically faster with the new approach Instead of only O(N 1/2 ) almost the trivial lower bound of O(N 1 ) is achieved as indicated by the black lines 1 Randomly select an initial set (ω i ) N 1 of directions 2 Construct the Voronoi tessellation (Ω i ) N 1 associated to the directions (ω i ) N 1 3 For each Voronoi region Ω i replace ω i by one of its mass centroidal directions 4 If not terminated then go to step 2 5 Compute the weights (B i ) N 1 by (3) The relaxation procedure is terminated if the maximum movement of all directions, ie the maximum angle of decentration, is below some threshold θ T For efficiency the Voronoi tessellation and the mass centroidal directions are approximated by uniformly sampling Ω Thus the actual Ω i implicitly are determined 6 by the samples closest to ω i The average of these closest points is the corresponding mass centroid Upon termination exactly these closest points are also used to approximate the uadrature weights B i This techniue in fact is very similar to k-means clustering as used in LightGen 1 This straightforward algorithm is very sensitive to the initial choice of (ω i ) N 1 and becoming trapped by local minima is almost unavoidable This is illustrated in the left image of figure 1, where far too many directions are located in unimportant regions This disadvantage easily can be overcome by incrementally determining the set of lighting directions The key is to insert a new direction nearby the direction ω i with the maximum weight B i : 1 Set n = 1 and select a random direction ω 2 Construct the Voronoi tessellation (Ω i ) n 1 associated to the directions (ω i ) n 1 3 For each Voronoi region Ω i replace ω i by one of its mass centroidal directions 4 If not terminated then go to step 2 5 Compute the weights (B i ) n 1 by (3) 6 If n < N increase n by 1 and choose a new direction ω n 1 nearby the direction ω i with the maximum weight B i and go to step 2 c The Eurographics Association 23

4 Kollig and Keller / Efficient Illumination by High Dynamic Range Images 31 Numerical Evidence The images in figure 1 illustrate the placement of N = 256 directions Clearly, the improved algorithm places more light sources in the brighter regions of the high dynamic range image The resulting uadrature rules have been used to render the images using (2) as shown in figure 2 While both uadrature rules are almost indistinguishable in unshadowed regions, the superiority of the improved uadrature rule becomes obvious in shadowed regions, which are rendered almost free of artifacts at already N = 256 directions The increased preprocessing time reliably is compensated by the uality gain This result can be explained by comparing both uadrature rules with respect to the maximum radiosity max i<n B i and the maximum angle of decentration after termination This angle is given by max i<n arccos ω i,γ i, where γ i is the centroidal direction of the Voronoi region corresponding to ω i As shown in figure 3, forcing more iterations by lowering the termination threshold θ T naturally decreases the maximum angle of decentration in the original approach In the improved version the maximum angle of decentration is marginally smaller at a somewhat higher variance At the same time the maximum radiosity practically remains unchanged independent of the approach However, the improved version almost achieves the trivial lower bound of O(N 1 ), which is uadratically better than the decay of the maximum radiosity in the original scheme As already mentioned at the end of section 2, this efficiently decreases the integration error of (2) 4 Efficient Anti-Aliasing Although our improved scheme results in much faster uadrature rules (see section 3), using an identical uadrature rule for the whole image results in clearly visible shadow boundary artifacts if the number N of light sources is small (see figure 2) Therefore two methods are proposed that allow one to reduce the total number of shadow rays while efficiently preventing aliasing Note that anti-aliasing is further improved by the powerful techniue of interleaved sampling 8, 1 However, the application of interleaved sampling to both methods proposed in the seuel is straightforward and therefore omitted for the sake of clarity 41 Randomly Perturbed Quadrature Rules At low sampling rates the shadow boundary artifacts can be turned into less objectionable noise by stratified sampling using the partition (Ω i ) N 1 However, for maximum efficiency the strata Ω i are approximated by inscribed spherical caps defined by cones of radius α i centered in ω i Thus it Figure 4: Interleaved uadrature rule consisting of M = 4 separate uadrature rules at N = 32 light sources The colored points represent the directions ω j,i of the light sources in the 2:1 lattitude/longitude map, while each color represents one uadrature rule Our improved scheme precisely captures the light distribution by the total of 128 = 4 32 directions as well as every separate uadrature rule does as for example highlighted by the solid points The image has been tone mapped for display is easy to generate a random direction inside each cone In order to decorrelate the samples this random perturbation of the uadrature rule has to be performed each time (2) is evaluated It is important to note that the variance of the original high dynamic range image L hdr restricted to the strata Ω i remains high due to the fine image details Therefore it has to be reduced, which is achieved by simply using the already filtered values B i 42 Interleaved Quadrature Rules Due to the correlation coefficient of the integrand with respect to the shadow rays 9 it pays off to send N shadow rays for a single primary ray, when computing the direct illumination Maximum efficiency is achieved by correlated sampling 9 We realize this kind of efficient multidimensional sampling for anti-aliasing by generating a separate uadrature rule ( ) N 1 ω j,i,b j,i for each primary ray j =,,M 1 and imposing the two constraints that both each separate uadrature rule ( ) N 1 ω j,i,b j,i for j =,,M 1 and ) M 1,N 1 the composite uadrature rule (ω j,i, B j,i j, fulfill the reuirements of section 2 Note that both constraints use the same directions, however, different uadrature weights B j,i and B j,i result from (3) as induced by the Voronoi tessellations of ( ) N 1 ω j,i for j =,,M 1 and ( ) M 1,N 1 ω j,i, respectively j, Figure 4 shows an example of an interleaved uadrature rule composited out of M = 4 separate uadrature rules with N = 32 light sources each that has been generated using the following algorithm: c The Eurographics Association 23

5 maximum angle of decentration Kollig and Keller / Efficient Illumination by High Dynamic Range Images number k of iterations maximum radiosity (weight) numer k of iterations Figure 5: Quality of the interleaved uadrature rule after k iterations The red lines indicate the maximum of the values of the M = 4 separate uadrature rules at N = 32 light sources, while the green lines indicate the values of the composite uadrature rule The unavoidable loss of uality of the separate uadrature rules is kept minimal by our generation process 1 Generate an initial uadrature rule ( ) N 1 ω,i,b,i following section 3 2 For j = 1,2,,M 1 construct the uadrature rule ( ) N 1 ω j,i,b j,i by randomly perturbing the directions ω,i 3 For j =,1,,M 1 construct the Voronoi tessellation (Ω j,i ) N 1 of Ω associated to the directions (ω j,i) N 1 4 For each Voronoi region Ω j,i replace ω j,i by one of its mass centroidal directions ) M 1,N 1 5 Construct the Voronoi tessellation ( Ω j,i of Ω j, associated to the directions (ω j,i ) M 1,N 1 j, 6 For each Voronoi region Ω j,i replace ω j,i by one of its mass centroidal directions 7 k times iterate steps 3 through 6 8 Compute the weights ( ) M 1,N 1 B j,i by (3) j, For the initialization step 2 the radius α of the cones of perturbation (see also section 41) is chosen small, eg α = 1 The alternate application of Lloyd s relaxation step to the separate uadrature rules in steps 3 and 4 and to the composite uadrature rule in steps 5 and 6 achieves to satisfy both constraints as mentioned in the beginning of this section Figure 5 shows the development of the uality of both the separate uadrature rules and the composite uadrature rule Upon initialization the maximum angle of decentration is small for the separate uadrature rules and due to a lack of correlation huge for the composite uadrature rule Then the iteration process efficiently increases correlation forcing a higher but nevertheless decreasing maximum angle of decentration in the separate uadrature rules In conseuence the maximum radiosities of the separate uadrature rules must be slightly increasing At the same time, however, the maximum radiosity of the composite uadrature rule remains small In practice only about k = 4 iteration steps are sufficient to obtain a high uality interleaved uadrature rule 5 Results Figure 6 compares rendering methods for computing the direct illumination by a high dynamic range image L hdr For ease of comparison the sunset 2:1 lattitude/longitude scan (see figures 1, 4, and 8) has been used The images on the left of figure 6 show the results of sampling the hemisphere (mid-left) and importance sampling the high dynamic range image (lower left) identifying each pixel as one directional light source The high variance of both estimators even cannot be sufficiently reduced by randomized uasi-monte Carlo 9 and conseuently strong noise remains visible However, as expected importance sampling the discrete set of light sources (lower left) is less noisy and avoids disturbing spike noise (see the enlargements in figure 6) The images on the right of figure 6 are rendered using our new techniues Obviously one single uadrature rule (see section 3) exposes clear shadow boundary artifacts (upper right), which are transferred to less perceivable noise (midright) by randomly perturbed uadrature rules (see section 41) Finally the interleaved uadrature rule (see section 42) reduces the shadow artifacts to an imperceptible level (lower right) While randomly perturbed uadrature rules may be favored for rendering still images, interleaved uadrature rules are the better choice for animations, where coherent sampling over time avoids flicker Although each method has used 32 samples for computing the direct illumination of each primary ray in our new rendering methods up to 15% of the shadow rays do not need to be shot, because they can be culled using the surface normal In addition better memory coherence and less pseudo random number generator calls are the reasons for up to 5% reduced rendering times Note that pseudo random number generation can consume considerable amounts of time, which also is the reason for the 25% increase of rendering time when randomly perturbc The Eurographics Association 23

6 Kollig and Keller / Efficient Illumination by High Dynamic Range Images a b c d e preprocessing time in s a b c d e rendering time in s a b c d e shadow rays per pixel (c) single uadrature rule (a) importance sampling of the hemisphere (d) randomly perturbed uadrature rule (b) importance sampling of the discrete set of light sources (e) interleaved uadrature rule Figure 6: Comparison of direct illumination computed by (a) importance sampling of the hemisphere, (b) importance sampling of the discrete set of light sources, (c) a single, (d) a randomly perturbed, and (e) an interleaved uadrature rule For antialiasing 16 primary rays per pixel were traced For each primary ray 32 samples were used for the techniues on the left The single and the randomly perturbed uadrature rule had N = 32 light sources and the interleaved uadrature rule consisted of M = 16 separate uadrature rules with N = 32 light sources each The diagram displays the preprocessing and rendering times on a 65 MHz Pentium III for the above images at a resolution of pixels and the number of shadow rays per pixel c The Eurographics Association 23

7 Kollig and Keller / Efficient Illumination by High Dynamic Range Images ing the single uadrature rule Considering the improved image uality at reduced rendering times the preprocessing of 23 seconds for a single or 75 seconds for an interleaved uadrature rule clearly pays off On the other hand obtaining the same uality without interleaved uadrature rules would reuire a single uadrature rule with much more light sources that in conseuence would take much longer time to generate 6 Conclusion We presented a robust algorithm for the efficient computation of direct illumination of predominantly diffuse surfaces by spherical high dynamic range images The scheme is designed for industrial production, where high resolution spherical high dynamic range scans are used (see figure 7) Without manual intervention our new scheme generates uadrature rules with minimized weights resulting in a considerable noise reduction and improved anti-aliasing As shown in figure 8 the light distribution is captured very precisely Thus it becomes redundant to eg cut out light sources manually 3 in order to reduce variance The resulting uadrature rules consume negligible amounts of memory and can be stored along with the high dynamic range images saving repeated preprocessing The directional light sources can be projected back onto a reconstructed geometry 2 thus allowing for even more precise shadowing An obvious improvement to our implementation is the acceleration of the computation of the mass centroids and the weights of the uadrature rules by graphics hardware Although specular surfaces are simple to render, the efficient combination with our techniues is not straightforward for general reflections properties, which cannot be represented as a weighted sum of basic bidirectional distribution functions This, as well as the generation of caustics, is subject to future research We also will focus on combining uadrature rules from multiple high dynamic range images and on enabling our scheme for illumination by high dynamic range video Acknowledgements The first author has been funded by the Stiftung Rheinland- Pfalz für Innovation The research has been supported by Spheron VR AG 15 References 1 Cohen, J, and Debevec, P "Light- Gen" HDRShop Plugin 21 jcohen/lightgen/lightgenhtml 1, 3 2 P Debevec Rendering Synthetic Objects Into Real Scenes: Bridging Traditional and Image-Based Graphics With Global Illumination and High Dynamic Range Photography In M Cohen, editor, SIGGRAPH 98 Conference Proceedings, Annual Conference Series, pages ACM SIGGRAPH, Addison-Wesley, July , 7 3 P Debevec, N Fong, and D Lemmon Image-Based Lighting In SIGGRAPH 22 Course Notes, number 5 ACM SIGGRAPH, July 22 1, 7 4 P Debevec and J Malik Recovering High Dynamic Range Radiance Maps from Photographs In T Whitted, editor, SIGGRAPH 97 Conference Proceedings, Annual Conference Series, pages ACM SIG- GRAPH, Addison-Wesley, August Q Du, V Faber, and M Gunzburger Centroidal Voronoi Tesselations: Applications and Algorithms SIAM Review, 41(4): , December S Hiller, O Deussen, and A Keller Tiled Blue Noise Samples In Proceedings of Vision, Modeling, and Visualization 21, pages IOS Press, H Jensen Importance Driven Path Tracing Using the Photon Map In P Hanrahan and W Purgathofer, editors, Rendering Techniues 95 (Proc 6th Eurographics Workshop on Rendering), pages Springer, A Keller and W Heidrich Interleaved Sampling In K Myszkowski and S Gortler, editors, Rendering Techniues 21 (Proc 12th Eurographics Workshop on Rendering), pages Springer, T Kollig and A Keller Efficient Multidimensional Sampling Computer Graphics Forum, 21(3): , September 22 4, 5 1 S Molnar Efficient Supersampling Antialiasing for High-Performance Architectures Technical Report TR91-23, The University of Noth Carolina at Chapel Hill, W Press, W Vetterling, S Teukolsky, and B Flannery Numerical Recipes in C++: the Art of Scientific Computing Cambridge University Press, 2nd edition, R Ramamoorthi and P Hanrahan An Efficient Representation for Irradiance Environment Maps In E Fiume, editor, Proceedings of ACM SIGGRAPH 21, Computer Graphics Proceedings, Annual Conference Series, pages ACM SIGGRAPH, August R Ramamoorthi and P Hanrahan Freuency Space Environment Map Rendering ACM Transaction on Graphics, 21(3): , July P Sloan, J Kautz, and J Snyder Precomputed Radiance Transfer for Real-Time Rendering in Dynamic, Low-Freuency Lighting Environments ACM Transaction on Graphics, 21(3): , July SpheronVR AG 1, 7 c The Eurographics Association 23

8 Kollig and Keller / Efficient Illumination by High Dynamic Range Images Figure 7: Reality check: A virtual car has been placed into a real set recorded by one high resolution high dynamic range image The scanned image is used both as background shot and for our fully automatic illumination techniue (a) single uadrature rule with N = 256 (c) single uadrature rule with N = 256 (b) interleaved uadrature rule with M = 8 and N = 32 (d) interleaved uadrature rule with M = 8 and N = 32 Figure 8: The top row example demonstrates our method for a predominantly continuous light distribution The Grace Cathedral example (courtesy P Debevec) in the bottom row impressively shows that our new method reliably captures bright regions and discontinuities of the high dynamic range image Note that no manual intervention like eg replacing the bright regions by area light sources (which would be even more tedious in the top row example) is reuired All images have been tone mapped for display c The Eurographics Association 23

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